Question 1: Palette 1/12

BITSAT 2025 MCQ

If ' $a$ ' is a complex number such that $|a|=1$. Find the value of $a$, so that the equation $a z^2+z+1=0$ has one purely imaginary root.

A

$\cos \left\{\cos ^{-1}\left(\frac{-\sqrt{5}+1}{4}\right)\right\}$

B

$\cos \left\{\sin ^{-1}\left(\frac{\sqrt{5}+1}{4}\right)\right\}+i \sin \left\{\cos ^{-1}\left(\frac{\sqrt{5}+1}{4}\right)\right\}$

C

$\sin \left\{\cos ^{-1}\left(\frac{\sqrt{5}-1}{4}\right)\right\}+i \sin ^{-1}\left(\frac{-\sqrt{5}+1}{2}\right)$

D

None of the above

Question 2: Palette 2/12

BITSAT 2025 MCQ

Let $z$ be a complex number for which $\left|2 z \cos \theta+z^2\right|>1$, if $|z|

A

equal to $\sqrt{2}-1$

B

greater than $\sqrt{2}+1$

C

less than $\sqrt{2}-1$

D

greater than $\sqrt{2}-1$

Question 3: Palette 3/12

BITSAT 2024 MCQ

The points represented by the complex number $ 1+i,-2+3 i, \frac{5}{3} i $ on the argand plane are

A

Vertices of an equilateral triangle

B

Vertical of an isosceles triangle

C

Collinear

D

None of the above

Question 4: Palette 4/12

BITSAT 2024 MCQ

The modulus of the complex number $ z $ such that $ |z+3-i|=1 $ and $ \arg (z)=\pi $ is equal to

A

3

B

2

C

9

D

4

Question 5: Palette 5/12

BITSAT 2023 MCQ

Number of solutions of the equation $z^2+|z|^2=0$ and $z \neq 0$ is

A

2

B

3

C

1

D

infinitely many solutions

Question 6: Palette 6/12

BITSAT 2023 MCQ

If $z_1$ and $z_2$ be nth root of unity which subtend a right angled at the origin. Then, $n$ must be of the form

A

$4 K+1$

B

$4 k$

C

$4 k+2$

D

$4 k+3$

Question 7: Palette 7/12

BITSAT 2022 MCQ

If $|w| = 2$, then the set of points $z = w - {1 \over w}$ is contained in or equal to the set of points z satisfying

A

$Im(z) = 0$

B

$|Im(z)| \le 1$

C

$|Re(z)| \le 2$

D

$|z| \le 3$

Question 8: Palette 8/12

BITSAT 2022 MCQ

The smallest positive integral value of n such that ${\left[ {{{1 + \sin {\pi \over 8} + i\cos {\pi \over 8}} \over {1 + \sin {\pi \over 8} - i\cos {\pi \over 8}}}} \right]^n}$ is purely imaginary, is equal to

A

4

B

3

C

2

D

8

Question 9: Palette 9/12

BITSAT 2021 MCQ

If Re(z + 2) = | z $-$ 2 |, then the locus of z is

A

parabola

B

circle

C

ellipse

D

hyperbola

Question 10: Palette 10/12

BITSAT 2020 MCQ

If $z = {{7 + i} \over {3 + 4i}}$, then z¹⁴ is

A

2⁷

B

2⁷i

C

($-$2)⁷

D

($-$2)⁷i

Question 11: Palette 11/12

BITSAT 2020 MCQ

The root of the equation $2(1 + i){x^2} - 4(2 - i)x - 5 - 3i = 0$, where $i = \sqrt { - 1} $, which has greater modulus, is

A

${{3 - 5i} \over 2}$

B

${{5 - 3i} \over 2}$

C

${{3 + i} \over 2}$

D

${{3i + 1} \over 2}$

Question 12: Palette 12/12

BITSAT 2020 MCQ

If $z = r{e^{i\theta }}$, then arg(e^iz) is

A

$-$r sin$\theta$

B

r cos$\theta$

C

e^{$-$r sin$\theta$}

D

$-$ r cos$\theta$